The Education Minister Richard Bruton says it is not in the country's interest to challenge the leadership of Enda Kenny.
Minister Bruton says now is not the time to undermine the Taoiseach and sap the energy of the Government.
It comes after the Government Chief Whip Regina Doherty, called on Mr Kenny to set out a timeline for his future as leader.
However, Mr Bruton believes now is not the time for the Taoiseach to go.
He said: "I believe very strongly that now is not the time to be challenging the leadership of Enda Kenny, it's not in the country's interest.
"Fine Gael has taken on the responsibility to confront the huge challenges in housing and health.
"In addition, we've just had the body blow of the British decision to leave the European Union. We need Enda Kenny's experience and relationships in Europe to steer us through this very difficult time."
Independent TD Finian McGrath says the Independent Alliance is looking to appoint somebody to mend fences with Fine Gael.
Mr McGrath said that the Cabinet is united to deal with the challenges ahead.
He said: "Brexit is a very important issue and we are all united in the Cabinet and the Independent Alliance are working very hard to deal with the major problems that could result from Brexit.
"At the moment we are trying to appoint a person who will try and mend the fences with Fine Gael, and that will be our objective over the next couple of days."